engineering formulas book

8/12/2019 Engineering Formulas Book

1/99

First Edition Volume 5

Formulas and Conversions


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Published by IDC Technologies,982 Wellington Street

WEST PERTH WA 6005WESTERN AUSTRALIACopyright 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004

IDC Technologies

A.B.N. 003 263 189

ISBN 1 875955 09 7

US English. First Edition.

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A Message from IDC TechnologiesTechnical Director,Steve Mackay

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Other books in this seriesVolume 1 INSTRUMENTATION

Automation using PLCs, SCADA and Telemetry, Process Control and

Data Acquisition

Volume 2 COMMUNICATIONS

Data Communications, Industrial Networking, TCP/IP and Fiber Optics

Volume 3 ELECTRICAL

Power Quality, Power Systems Protection and Substation Automation

Volume 4 ELECTRONICS

Personal Computers, Digital Signal Processing and Analog/Digital Conversions


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Table of Contents

Chapter 1Definition and Abbreviations for Physical Quantities ...........1

Chapter 2

Units of Physical Quantities .................................................3

Chapter 3System of Units ..................................................................23

Chapter 4

General Mathematical Formulae........................................274.1 Algebra .................................................................................. 274.2 Geometry...............................................................................294.3 Trigonometry ......................................................................... 39

4.4 Logarithm ..............................................................................404.5 Exponents .............................................................................424.6 Complex Numbers ................................................................ 42

Chapter 5

Engineering Concepts and Formulae ................................445.1 Electricity ............................................................................... 445.2 Applied Mechanics ................................................................ 57

5.2.1 Newton's laws of motion........................................................... 575.2.2 Linear Velocity And Acceleration ............................................. 605.2.3 Force ........................................................................................ 615.2.4 Centripetal (Centrifugal) Force................................................. 625.2.5 Stress, Strain And Modulus Of Elasticity ................................. 64

5.3 Thermodynamics................................................................... 645.3.1 Laws of Thermodynamics ........................................................ 645.3.2 Momentum ............................................................................... 655.3.3 Impulse..................................................................................... 655.3.4 Elastic and Inelastic collision.................................................... 655.3.5 Center of Mass ......................................................................... 65

5.3.6 Angular Motion ......................................................................... 65


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5.3.7 Conditions of Equilibrium ......................................................... 655.3.8 Gravity ...................................................................................... 665.3.9 Vibrations & Waves.................................................................. 665.3.10 Standing Waves ....................................................................... 665.3.11 Beats ........................................................................................ 665.3.12 Temperature and Heat ............................................................. 675.3.13 Ideal Gases .............................................................................. 675.3.14 Elastic Deformation.................................................................. 685.3.15 Temperature Scales................................................................. 685.3.16 Sensible Heat Equation............................................................ 685.3.17 Latent Heat ............................................................................... 685.3.18 Gas Laws ................................................................................. 685.3.19 Specific Heats Of Gases .......................................................... 695.3.20 Efficiency of Heat Engines ....................................................... 705.3.21 Heat Transfer by Conduction ................................................... 715.3.22 Thermal Expansion of Solids ................................................... 725.3.23 Chemical Heating Value of a Fuel ........................................... 72

5.4 Fluid Mechanics ....................................................................775.4.1 Discharge from an Orifice ........................................................ 775.4.2 Bernoullis Theory..................................................................... 785.4.3 Actual pipe dimensions ............................................................ 78

Chapter 6

References.........................................................................806.1 Periodic Table of Elements ...................................................806.2 Resistor Color Coding ...........................................................81


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Definition and Abbreviations for Physical Quantities

Symbol Unit Quantity

m meter Length

kg kilogram Mass

s second Time

A ampere Electric current

K kelvin Thermodynamic temp

cd candela Luminous intensity

Quantity Unit Symbol Equivalent

Plane angle radian rad -

Force newton N kg m/s2

Work, energy heat joule JNm

Power watt W J/s

Frequency hertz Hz s-1

Viscosity:kinematic

- m2/s 10 c St(Centistoke)

Viscosity:Dynamic

- Ns/m2 103cP(Centipoise)

Pressure - Pa or N/m2 pascal, Pa

Symbol Prefix Factor by which unit ismultiplied

T Tera 1012

G Giga 109

M Mega 106

Chapter 1


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Symbol Prefix Factor by which unit ismultiplied

k Kilo 103

h Hecto 102

da Deca 10

d Deci 10-1

c Centi 10-2

m Milli 10-3

Micro 10-6

n Nano 10-9

p Pico 10-12

Quantity Electricalunit

Symbol Derivedunit

Potential Volt V W/A

Resistance Ohm V/A

Charge Coulomb C As

Capacitance Farad F As/V

Electric fieldstrength

- V/m -

Electric flux

density

- C/m2 -

Quantity Magnetic

unit

Symbol Derived unit

Magnetic flux Weber Wb Vs = Nm/A

Inductance Henry H Vs/A = Nm/A2

Magnetic field

strength

- A/m -

Magnetic flux density Tesla T Wb/m2=

(N)/(Am)


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Units of Physical Quantities

Conversion Factors (general):

1 acre = 43,560 square feet

1 cubic foot = 7.5 gallons

1 foot = 0.305 meters

1 gallon = 3.79 liters

1 gallon = 8.34 pounds

1 grain per gallon = 17.1 mg/L

1 horsepower = 0.746 kilowatts

1 million gallons per day = 694 gallons per minute

1 pound = 0.454 kilograms

1 pound per square inch = 2.31 feet of water

Degrees Celsius = (Degrees Fahrenheit - 32) (5/9)

Degrees Fahrenheit = (Degrees Celsius) (9/5) + 32

1% = 10,000 mg/L

Name To convert from ToMultiply

byDivide by

Acceleration ft/sec2 m/s2 0.3048 3.2810

Area acre m2 4047 2.471E-04

Area ft2 m2 9.294E-02 10.7600

Area hectare m2 1.000E+04 1.000E-04

Area in2 m2 6.452E-04 1550

Density g/cm3 kg/m3 1000 1.000E-03

Density lbm/ft3 kg/m3 16.02 6.243E-02

Density lbm/in3 kg/m3 2.767E+04 3.614E-05

Chapter 2


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Name To convert from ToMultiplyby

Divide by

Density lbs2/in4 kg/m3 1.069E+07 9.357E-08

Density slug/ft3 kg/m3 515.40 1.940E-03

Energy BTU J 1055 9.478E-04

Energy cal J 4.1859 0.2389

Energy erg J 1.000E-07 1.000E+07

Energy eV J 1.602E-19 6.242E+18

Energy Ftlbf J 1.3557 0.7376

Energy kiloton TNT J 4.187E+12 2.388E-13

Energy KWhr J 3.600E+06 2.778E-07

Energy Megaton TNT J 4.187E+15 2.388E-16

Force Dyne N 1.000E-05 1.000E+05

Force Lbf N 4.4484 0.2248

Force Ozf N 0.2780 3.5968

Heat capacity BTU/lbm F J/kgC 4188 2.388E-04

Heat transfer coefficient BTU/hrft2F W/m2C 5.6786 0.1761

Length AU m 1.496E+11 6.685E-12

Length ft m 0.3048 3.2810

Length in m 2.540E-02 39.3700

Length mile m 1609 6.214E-04

Length Nautical mile m 1853 5.397E-04

Length parsec m 3.085E+16 3.241E-17

Mass amu kg 1.661E-27 6.022E+26

Mass lbm kg 0.4535 2.2050

Mass lbs2/in kg 1200.00 5.711E-03

Mass slug kg 14.59 6.853E-02

Mass flow rate lbm/hr kg/s 1.260E-04 7937


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Divide by

Mass flow rate lbm/sec kg/s 0.4535 2.2050

Moment of inertia ftlbs2 kgm2 1.3557 0.7376

Moment of inertia inlbs2 kgm2 0.1130 8.8510

Moment of inertia ozins2 kgm2 7.062E-03 141.60

Power BTU/hr W 0.2931 3.4120

Power hp W 745.71 1.341E-03

Power tons of refrigeration W 3516 2.844E-04

Pressure bar Pa 1.000E+05 1.000E-05

Pressure dyne/cm2

Pa 0.1000 10.0000

Pressure in. mercury Pa 3377 2.961E-04

Pressure in. water Pa 248.82 4.019E-03

Pressure kgf/cm2 Pa 9.807E+04 1.020E-05

Pressure lbf/ft2 Pa 47.89 2.088E-02

Pressure lbf/in2 Pa 6897 1.450E-04

Pressure mbar Pa 100.00 1.000E-02

Pressure microns mercury Pa 0.1333 7.501

Pressure mm mercury Pa 133.3 7.501E-03

Pressure std atm Pa 1.013E+05 9.869E-06

Specific heat BTU/lbmF J/kgC 4186 2.389E-04

Specific heat cal/gC J/kgC 4186 2.389E-04

Temperature F C 0.5556 1.8000

Thermal conductivity BTU/hrftF W/mC 1.7307 0.5778

Thermal conductivity BTUin/hrft2F W/mC 0.1442 6.9340

Thermal conductivity cal/cmsC W/mC 418.60 2.389E-03

Thermal conductivity cal/fthrF W/mC 6.867E-03 145.62

Time day S 8.640E+04 1.157E-05


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Divide by

Time sidereal year S 3.156E+07 3.169E-08

Torque ftlbf Nm 1.3557 0.7376

Torque inlbf Nm 0.1130 8.8504

Torque Inozf Nm 7.062E-03 141.61

Velocity ft/min m/s 5.079E-03 196.90

Velocity ft/s m/s 0.3048 3.2810

Velocity Km/hr m/s 0.2778 3.6000

Velocity miles/hr m/s 0.4470 2.2370

Viscosity absolute centipose Ns/m2

1.000E-03 1000

Viscosity absolute g/cms Ns/m2 0.1000 10

Viscosity absolute lbf/ft2s Ns/m2 47.87 2.089E-02

Viscosity absolute lbm/fts Ns/m2 1.4881 0.6720

Viscosity kinematic centistoke m2/s 1.000E-06 1.000E+06

Viscosity kinematic ft2/sec m2/s 9.294E-02 10.7600

Volume ft3 m3 2.831E-02 35.3200

Volume in3 m3 1.639E-05 6.102E+04

Volume Liters m3 1.000E-03 1000

Volume U.S. gallons m3 3.785E-03 264.20

Volume flow rate ft3/min m3/s 4.719E-04 2119

Volume flow rate U.S. gallons/min m3/s 6.309E-05 1.585E+04

A. DISTANCE (Length)Conversions

Multiply By To obtain

LENGTH

Centimeter 0.03280840 foot

Centimeter 0.3937008 inch


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To Convert To Multiply By

Kilometers Statute Miles 0.621371192

Kilometers Meters 1000

Leagues, Nautical Nautical Miles 3

Leagues, Nautical Kilometers 5.556

Leagues, Statute Statute Miles 3

Leagues, Statute Kilometers 4.828032

Links, (Surveyor's) Chains 0.01

Links, (Surveyor's) Inches 7.92

Links, (Surveyor's) Centimeters 20.1168

Meters Statute Miles 0.000621371

Meters Kilometers 0.001

Meters Yards 1.093613298

Meters Feet 3.280839895

Meters Inches 39.370079

Meters Centimeters 100

Meters Millimeters 1000

Microns Meters 0.000001

Microns Inches 0.0000394

Miles, Nautical Statute Miles 1.1507794

Miles, Nautical Kilometers 1.852

Miles, Statute Kilometers 1.609344

Miles, Statute Furlongs 8

Miles, Statute Rods 320

Miles, Statute Meters 1609.344

Miles, Statute Yards 1760

Miles, Statute Feet 5280

Miles, Statute Inches 63360


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Miles, Statute Centimeters 160934.4

Millimeters Inches 0.039370079

Mils Inches 0.001

Mils Millimeters 0.0254

Paces (US) Inches 30

Paces (US) Centimeters 76.2

Points (Typographical) Inches 0.013837

Points (Typographical) Millimeters 0.3514598

Rods Meters 5.0292

Rods Yards 5.5

Rods Feet 16.5

Spans Inches 9

Spans Centimeters 22.86

Yards Miles 0.00056818

Yards Meters 0.9144

Yards Feet 3

Yards Inches 36

Yards Centimeters 91.44

Conversion

Length

1 ft = 12 in 1 yd = 3 ft

1 cm = 0.3937 in 1 in = 2.5400 cm

1 m = 3.281 ft 1 ft = 0.3048 m

1 m = 1.0936 yd 1 yd = 0.9144 m

1 km = 0.6214 mile 1 mile = 1.6093 km

1 furlong = 40 rods 1 fathom = 6 ft


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Conversion

1 statute mile = 8 furlongs 1 rod = 5.5 yd

1 statute mile = 5280 ft 1 in = 100 mils

1 nautical mile = 6076 ft 1 light year = 9.461 x 1015m

1 league = 3 miles 1 mil = 2.540 x 10-5m

Area

1 ft2 = 144 in2 1 acre = 160 rod2

1 yd2 = 9 ft2 1 acre = 43,560 ft2

1 rod2 = 30.25 yd2 1 mile2 = 640 acres

1 cm2 = 0.1550 in2 1 in2 = 6.4516 cm2

1 m2 = 10.764 ft2 1 ft2 = 0.0929 m2

1 km2 = 0.3861 mile2 1 mile2 = 2.590 km2

Volume

1 cm3 = 0.06102 in3 1 in3 = 16.387 cm3

1 m3 = 35.31 ft3 1 ft3 = 0.02832 m3

1 Litre = 61.024 in3 1 in3 = 0.0164 litre

1 Litre = 0.0353 ft3 1 ft3 = 28.32 litres

1 Litre = 0.2642 gal. (U.S.) 1 yd3 = 0.7646 m3

1 Litre = 0.0284 bu (U.S.) 1 gallon (US) = 3.785 litres

1 Litre = 1000.000 cm3 1 gallon (US) = 3.785 x 10-3m3

1 Litre = 1.0567 qt. (liquid) or0.9081 qt. (dry)

1 bushel (US) = 35.24 litres

1 oz (US fluid) = 2.957 x 10-5m3 1 stere = 1 m3

Liquid Volume

1 gill = 4 fluid ounces 1 barrel = 31.5 gallons

1 pint = 4 gills 1 hogshead = 2 bbl (63 gal)

1 quart = 2 pints 1 tun = 252 gallons

1 gallon = 4 quarts 1 barrel (petrolum) = 42 gallons


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Conversion

Dry Volume

1 quart = 2 pints 1 quart = 67.2 in3

1 peck = 8 quarts 1 peck = 537.6 in3

1 bushel = 4 pecks 1 bushel = 2150.5 in3

B. AreaConversions


AREA

acre 4,046.856 meter2

(m2

)

acre 0.4046856 hectare

centimeter2 0.1550003 inch2

centimeter2 0.001076391 foot2

foot2 0.09290304* meter2(m2)

foot2 929.03042 centimeter2(cm2)

foot2 92,903.04 millimeter2(mm2)

hectare 2.471054 acre

inch2 645.16* millimeter2(mm2)

inch2 6.4516 centimeter2(cm2)

inch2 0.00064516 meter2(m2)

meter2 1,550.003 inch2

meter2 10.763910 foot2

meter2 1.195990 yard2

meter2 0.0002471054 acre

millimeter2 0.00001076391 foot2

millimeter2 0.001550003 inch2

yard2 0.8361274 meter2(m2)


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C. VolumeConversionsMetric Conversion Factors: Volume (including Capacity)


VOLUME (including CAPACITY)

centimeter3 0.06102376 inch3

foot3 0.028311685 meter3(m3)

foot3 28.31685 liter

gallon (UK liquid) 0.004546092 meter3(m3)

gallon (UK liquid) 4.546092 litre

gallon (US liquid) 0.003785412 meter3(m3)

gallon (US liquid) 3.785412 liter

inch3 16,387.06 millimeter3(mm3)

inch3 16.38706 centimeter3(cm3)

inch3 0.00001638706 meter3(m3)

Liter 0.001* meter3(m3)

Liter 0.2199692 gallon (UK liquid)

Liter 0.2641720 gallon (US liquid)

Liter 0.03531466 foot3

meter3 219.9692 gallon (UK liquid)

meter3 264.1720 gallon (US liquid)

meter3 35.31466 foot3

meter3 1.307951 yard3

meter3 1000.* liter

meter3 61,023.76 inch3

millimeter3 0.00006102376 inch3

Yard3 0.7645549 meter3(m3)

D. Mass and WeightConversions


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Carat Milligrams 200

Drams, Avoirdupois Avoirdupois Ounces 0.06255

Drams, Avoirdupois Grams 1.7718452

Drams, Avoirdupois Grains 27.344

Drams, Troy Troy Ounces 0.125

Drams, Troy Scruples 3

Drams, Troy Grams 3.8879346

Drams, Troy Grains 60

Grains Kilograms 6.47989E-05

Grains Avoirdupois Pounds 0.00014286

Grains Troy Pounds 0.00017361

Grains Troy Ounces 0.00208333

Grains Avoirdupois Ounces 0.00228571

Grains Troy Drams 0.0166

Grains Avoirdupois Drams 0.03657143

Grains Pennyweights 0.042

Grains Scruples 0.05

Grains Grams 0.06479891

Grains Milligrams 64.79891

Grams Kilograms 0.001

Grams Avoirdupois Pounds 0.002204623

Grams Troy Pounds 0.00267923

Grams Troy Ounces 0.032150747

Grams Avoirdupois Ounces 0.035273961

Grams Avoirdupois Drams 0.56438339

Grams Grains 15.432361


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Pounds, Troy Troy Ounces 12

Pounds, Troy Avoirdupois Ounces 13.16571

Pounds, Troy Avoirdupois Drams 210.6514

Pounds, Troy Pennyweights 240

Pounds, Troy Grams 373.2417216

Pounds, Troy Grains 5760

Quintals Metric Tons 0.1

Quintals Kilograms 100

Quintals Avoirdupois Pounds 220.46226

Scruples Troy Drams 0.333

Scruples Grams 1.2959782

Scruples Grains 20

Tons, Long (Deadweight) Metric Tons 1.016046909

Tons, Long (Deadweight) Short Tons 1.12

Tons, Long (Deadweight) Long Hundredweights 20

Tons, Long (Deadweight) Short Hundredweights 22.4

Tons, Long (Deadweight) Kilograms 1016.04691

Tons, Long (Deadweight) Avoirdupois Pounds 2240

Tons, Long (Deadweight) Avoirdupois Ounces 35840

Tons, Metric Long Tons 0.9842065

Tons, Metric Short Tons 1.1023113

Tons, Metric Quintals 10

Tons, Metric Long Hundredweights 19.68413072

Tons, Metric Short Hundredweights 22.04623

Tons, Metric Kilograms 1000

Tons, Metric Avoirdupois Pounds 2204.623

Tons, Metric Troy Ounces 32150.75


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Tons, Short Long Tons 0.8928571

Tons, Short Metric Tons 0.90718474

Tons, Short Long Hundredweights 17.85714

Tons, Short Short Hundredweights 20

Tons, Short Kilograms 907.18474

Tons, Short Avoirdupois Pounds 2000

E. DensityConversions


Grains/imp. Gallon Parts/million 14.286

Grains/US gallon Parts/million 17.118

Grains/US gallon Pounds/million gal 142.86

Grams/cu. Cm Pounds/mil-foot 3.405E-07

Grams/cu. Cm Pounds/cu. in 0.03613

Grams/cu. Cm Pounds/cu. ft 62.43

Grams/liter Pounds/cu. ft 0.062427

Grams/liter Pounds/1000 gal 8.345

Grams/liter Grains/gal 58.417

Grams/liter Parts/million 1000

Kilograms/cu meter Pounds/mil-foot 3.405E-10

Kilograms/cu meter Pounds/cu in 0.00003613

Kilograms/cu meter Grams/cu cm 0.001

Kilograms/cu meter Pound/cu ft 0.06243

Milligrams/liter Parts/million 1

Pounds/cu ft Pounds/mil-foot 5.456E-09

Pounds/cu ft Pounds/cu in 0.0005787


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Pounds/cu ft Grams/cu cm 0.01602

Pounds/cu ft Kgs/cu meter 16.02

Pounds/cu in Pounds/mil-foot 0.000009425

Pounds/cu in Gms/cu cm 27.68

Pounds/cu in Pounds/cu ft 1728

Pounds/cu in Kgs/cu meter 27680

F. Relative Density (Specific Gravity) Of Various Substances

Substance Relative

Density

Water (fresh) 1.00

Mica 2.9

Water (sea average) 1.03

Nickel 8.6

Aluminum 2.56

Oil (linseed) 0.94

Antimony 6.70

Oil (olive) 0.92

Bismuth 9.80

Oil (petroleum) 0.76-0.86

Brass 8.40

Oil (turpentine) 0.87

Brick 2.1

Paraffin 0.86

Calcium 1.58

Platinum 21.5

Carbon (diamond) 3.4


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Substance RelativeDensity

Sand (dry) 1.42

Carbon (graphite) 2.3

Silicon 2.6

Carbon (charcoal) 1.8

Silver 10.57

Chromium 6.5

Slate 2.1-2.8

Clay 1.9

Sodium 0.97

Coal 1.36-1.4

Steel (mild) 7.87

Cobalt 8.6

Sulphur 2.07

Copper 8.77

Tin 7.3

Cork 0.24

Tungsten 19.1

Glass (crown) 2.5

Wood (ash) 0.75

Glass (flint) 3.5

Wood (beech) 0.7-0.8

Gold 19.3

Wood (ebony) 1.1-1.2

Iron (cast) 7.21

Wood (elm) 0.66

Iron (wrought) 7.78


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Substance RelativeDensity

Wood (lignum-vitae) 1.3

Lead 11.4

Magnesium 1.74

Manganese 8.0

Mercury 13.6

Lead 11.4

Magnesium 1.74

Manganese 8.0

Wood (oak) 0.7-1.0

Wood (pine) 0.56

Wood (teak) 0.8

Zinc 7.0

Wood (oak) 0.7-1.0

Wood (pine) 0.56

Wood (teak) 0.8

Zinc 7.0

Mercury 13.6

G. Greek Alphabet

NameLowerCase

UpperCase

Alpha

Beta

Gamma

Delta

Epsilon

Zeta


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NameLowerCase

UpperCase

Eta

Theta

Iota

Kappa

Lambda

Mu

Nu

Xi

Omicron

Pi

Rho

Sigma and

Tau

Upsilon

Phi

Chi

Psi

Omega


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System of UnitsThe two most commonly used systems of units are as follows:

SIImperial

SI:The International System of Units (abbreviated "SI") is a scientific method of expressing

the magnitudes of physical quantities. This system was formerly called the meter-kilogram-

second (MKS) system.

Imperial:A unit of measure for capacity officially adopted in the British Imperial System;

British units are both dry and wet

Metric System

Exponent

value

Numerical

equivalentRepresentation Example

Tera 1012 1000000000000 TThz (Tera

hertz)

Giga 109 1000000000 GGhz (Giga

hertz)

Mega 106 1000000 MMhz (Mega

hertz)

Unitquantity

1 1hz (hertz)F (Farads)

Micro 10-6 0.001 F (Microfarads)

Nano 10-9 0.000001 nnF (Nano

farads)

Pico 10-12 0.000000000001 ppF (Pico

farads)

Conversion Chart

Mu l t i p l y

b y

IntoMilli

IntoCenti

IntoDeci

IntoMGL*

IntoDeca

IntoHecto

IntoKilo

To

convert

Kilo

106 105 104 103 102 101 1

Chapter 3


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- 24 -

Mu l t i p l y

b y

IntoMilli

IntoCenti

IntoDeci

IntoMGL*

IntoDeca

IntoHecto

IntoKilo

To

convert

Hecto

105 104 103 102 101 1 10-1

Toconvert

Deca

104 103 102 101 1 10-1 10-2

To

convertMGL*

103 102 101 1 10-1 10-2 10-3

To

convertDeci

102 101 1 10-1 10-2 10-3 10-4

To

convertCenti 101

1 10-1

10-2

10-3

10-4

10-5

To

convertMilli

1 10-1 10-2 10-3 10-4 10-5 10-6

MGL = meter, gram, liter

Example:To convert Kilogram Into Milligram (1 Kilo X 106) Milligrams

Physical constants

NameSymbolic

RepresentationNumerical Equivalent

Avogadro's number N 6.023 x 1026/(kg mol)

Bohr magneton B 9.27 x 10-24Am 252

Boltzmann's constant k 1.380 x 10-23J/k

Stefan-Boltzmann constant d 5.67 x 10-8W/(m2K4)

Characteristic impedance of freespace

Zo (o/Eo)1/2=120

Electron volt eV 1.602 x 10-19 J

Electron charge e 1.602 x 10-19 C


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NameSymbolic


Electronic rest mass me 9.109 x 10-31 kg

Electronic charge to mass ratio e/me 1.759 x 10

11

C/kg

Faraday constant F 9.65 x 107C/(kg mol)

Permeability of free space 0 4x 10-7H/m

Permittivity of free space Eo 8.85 x 10-12F/m

Planck's constant h 6.626 x 10-34J s

Proton mass mp 1.672 x 10-27kg

Proton to electron mass ratio mp/me 1835.6

Standard gravitational

accelerationg 9.80665 m/s2, 9.80665 N/kg

Universal constant of gravitation G 6.67 x 10-11 N m2/kg2

Universal gas constant Ro 8.314 kJ/(kg mol K)

Velocity of light in vacuum C 2.9979 x 108m/s

Temperature 0C 5/9(0F - 32)

Temperature K5/9(0F + 459.67), 5/90R, 0C +

273.15

Speed of light in air c 3.00 x 108m s-1

Electron charge e -1.60 x 10-19C

Mass of electron me 9.11 x 10-31kg

Planck's constant h 6.63 x 10-34J s

Universal gravitational constant G 6.67 x 10-11N m2kg-2

Electron volt 1 eV 1.60 x 10-19J

Mass of proton mp 1.67 x 10-27kg


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NameSymbolic


Acceleration due to gravity onEarth

g 9.80 m s-2

Acceleration due to gravity on the

Moon gM 1.62 m s-2

Radius of the Earth RE 6.37 x 106m

Mass of the Earth ME 5.98 x 1024kg

Radius of the Sun RS 6.96 x 108m

Mass of the Sun MS 1.99 x 1030kg

Radius of the Moon RM 1.74 x 106m

Mass of the Moon MM 7.35 x 1022

kg

Earth-Moon distance - 3.84 x 108m

Earth-Sun distance - 1.50 x 1011m

Speed of light in air c 3.00 x 108m s-1

Electron charge e -1.60 x 10-19C

Mass of electron me 9.11 x 10-31kg

Planck's constant h 6.63 x 10

-34

J s

Universal gravitational constant G 6.67 x 10-11N m2kg-2

Electron volt 1 eV 1.60 x 10-19J

Mass of proton mp 1.67 x 10-27kg

Acceleration due to gravity onEarth

g 9.80 m s-2

Acceleration due to gravity on theMoon

gM 1.62 m s-2

Ton 1 ton 1.00 x 103kg


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General Mathematical Formulae

4.1 AlgebraA. Expansion FormulaeSquare of summation

(x + y)2= x2+ 2xy + y2

Square of difference

(x y)2= x2 2xy + y2

Difference of squares

x2 y2= (x + y) (x y)

Cube of summation

(x + y)3= x3+ 3x2y + 3xy2+ y3

Summation of two cubes

x3+ y3= (x + y) (x2- xy + y2)

Cube of difference

(x y)3= x3 3x2y + 3xy2 y3

Difference of two cubes

x3 y3= (x y) (x2+ xy + y2)

B. Quadratic Equation

If ax2+ bx + c = 0,

Then

2 4

2

b b acx

a

=

The basic algebraic properties of real numbers a, b and c are:

Property Description

Closure a + b and ab are real numbers

Commutative a + b = b + a, ab = ba

Associative (a+b) + c = a + (b+c), (ab)c = a(bc)

Distributive (a+b)c = ac+bc

Chapter 4


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Identity a+0 = 0+a = a

Inverse a + (-a) = 0, a(1/a) = 1

Cancellation If a+x=a+y, then x=y

Zero-factor a0 = 0a = 0

Negation -(-a) = a, (-a)b= a(-b) = -(ab), (-a)(-b) = ab

Algebraic Combinations

Factors with a common denominator can be expanded:

c

b

c

a

c

ba+=

+

Fractions can be added by finding a common denominator:

cd

bcad

d

b

c

a +=+

Products of fractions can be carried out directly:

cd

ab

d

b

c

a=

Quotients of fractions can be evaluated by inverting and multiplying:

bc

ad

c

d

b

a

dc

ba ==

Radical Combinations

nnn baab=

nn aa /1=

n

nn

b

a

b

a =

n

m

n m aa =

mnn m aa =


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4.2 Geometry

Item

Circumference

/ Perimeter Area Surface Area Volume

Square 4s s2 NA NA

Rectangle 2 (L + B)(Length)(Breadth)

= LBNA NA


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ItemCircumference

/ PerimeterArea Surface Area Volume

Triangle

s1+ s2+ s3where s1,s2,s3are the 3 sidesof the triangle

HB 2

1 NA NA

Right

triangle

s1+ s2+ s3 HB 2

1 NA NA


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ItemCircumference


Generic

triangle

s1+ s2+ s3

))()(( csbsass

where

2

cbas

++= NA NA

Equilateral

triangle

3s

where s is the

length of eachside

bhA2

1= NA NA


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ItemCircumference


Trapezoidwhere and arethe 2 base angles

hba

A

+=2

NA NA

Circle

C = 2rC = d

A = r2

NA NA


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ItemCircumference


Circle

Sector

2r + (arc

length)

2

rarcA

=

2

360rA

=

2

2rA

=

NA NA

Ellipse

(1/4)Dd

where D and dare the two axis

DdA4=

D is the larger

radius and d is thesmaller radius

NA NA


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ItemCircumference


Trapezoid Sum of all sides hbbA )(2

121+= NA NA

Hexagon 6s A = 2.6s2

Where s is the

length of 1 side

NA NA


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ItemCircumference


Octagon 8sA = 4.83 s2

Where s is thelength of 1 side

NA NA

Cube NA NA 6s2 s

3


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ItemCircumference


Rectangular

solidNA NA

2 l h + 2wh + 2 l w h

Right

cylinderNA NA

S = 2rh +2r2 V = r2h


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ItemCircumference


Sphere NA NAS = 4r2

3

4r3

Pyramid NA NA.perimeterslant height +

B

3

1base area

perpendicularheight


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ItemCircumference


Rectangular

prismNA NA 2lh+2lw+2wh V = lwh

Cone NA NA pir(r+sh) 3

1r2h


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4.3 Trigonometry

A. Pythagoras' Law

c2= a

2+ b

2

B. Basic RatiosSin = a/cCos = b/cTan = a/bCosec = c/aSec = c/bCot = b/a

Degrees versus Radians

A circle in degree contains 360 degreesA circle in radians contains 2radians

Sine, Cosine and Tangent

sin opposite

hypotenus= cos

adjacent

hypotenus= tan

opposite

adjacent=

Sine, Cosine and the Pythagorean Triangle

[ ] [ ]2 2 2 2sin cos 1sin cos + = + =

ca

b

opposite

adjacent

hypotenuse


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Tangent, Secant and Co-Secant

sintan

cos

=

1sec cos =

1csc

sin

=

C. Trigonometric Function Values

Eulers Representation

cos( ) sin( )j

e j = +

cos( ) sin( )j

e j =

cos( ) sin( )jne n j n = +

cos2

j je e

+=

sin 2

j je e

j

=

4.4 Logarithm

Definition

The logarithm of a number to a particular base is the power (or index)to which that

basemust be raised to obtain the number.

The number 8 written in index formas 8 = 23

The equation can be rewritten in logarithm formas2log 8 = 3

Logarithm laws

The logarithm laws are obtained from the index laws and are:

logax + logay = logaxy


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logax logay = loga(x/y)

logaxy = y logax

loga(1/x) = -logax

loga1 = 0

logaa = 1

xa xa =)(log

Note:It is not possible to have the logarithm of a negative number. All logarithms must

have the same base.

Euler RelationshipThe trigonometric functions are related to a complex exponential by the Eulerrelationship:

xjxejx sincos +=

xjxe jx sincos = From these relationships the trig functions can be expressed in terms of the complex

exponential:

2cos

jxjx eex

+=

2sin

jxjx eex

=

Hyperbolic Functions

The hyperbolic functions can be defined in terms of exponentials.

Hyperbolic sine = sinh x =2

xx ee

Hyperbolic cosine = cosh x =2

xx ee +

Hyperbolic tangent = tanh x =xx

xx

ee

ee

x

x

+

=

cosh

sinh


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4.5 ExponentsSummary of the Laws of Exponents

Let c, d, r, and sbe any real numbers.

srsr ccc += rrr dcdc = )(

0= c,cc

c srs

r

0=

d,d

c

d

cr

rr

srsrcc

=)( r

r

cc

1=

Basic Combinations

Since the raising of a number n to a power p may be defined as multiplying

n times itself p times, it follows that

2121 pppp nnn =+

The rule for raising a power to a power can also be deduced

(na)b= nab

(ab)n = anbn

am/an= am-n

where a not equal to zero

4.6 Complex NumbersA complex number is a number with a real and an imaginary part, usuallyexpressed in Cartesian form

a + jb where j = -1 and j j = -1

Complex numbers can also be expressed in polar form

Aej

where A = a2+b2and = tan-1(b/a)

The polar form can also be expressed in terms of trigonometric functions using the Euler

relationshipej

= cos + j sin

Euler Relationship

The trigonometric functions are related to a complex exponential by the

Euler relationship

ejx = cos x + j sin x


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e-j

= cos x - j sin x

From these relationships the trigonometric functions can be expressed in terms of the

complex exponential:

2cos

jxjxee

x+

=

2sin

jxjxee

x

=

This relationship is useful for expressing complex numbers in polar form, as

well as many other applications.

Polar Form, Complex Numbers

The standard form of a complex number is

a + jb where j = -1

But this can be shown to be equivalent to the form

Aej

where A = a2+b2and = tan-1(b/a)

which is called the polar form of a complex number. The equivalence can be shown byusing the Euler relationship for complex exponentials.

)tansintan(cos 1122

+

+= a

bj

a

bbaAe

j

jbaba

bj

ba

abaAe

j +=+

++

+= )(2222

22


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Engineering Concepts and Formulae

5.1 Electricity

Ohm's Law

R

VI=

Or

V = IR

WhereI = current (amperes)E = electromotive force (volts)R = resistance (ohms)

Temperature correction

Rt= Ro (1 + t)

WhereRo = resistance at 0C (.)

Rt= resistance at tC (.)= temperature coefficient which has an average value for copper of 0.00428 (/C)

)1(

)1(

1

212

t

tRR

++

=

Where R1= resistance at t1

R2= resistance at t2

Values of

alpha

/C

Copper 0.00428

Platinum 0.00358

Nickel 0.00672

Tungsten 0.00450

Chapter 5


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Aluminum 0.0040

Current, nqvAt

nqvtAI ==

Conductor Resistivity

a

LR

=

Where

= specific resistance (or resistivity) (ohm meters, m)L = length (meters)

a = area of cross-section (square meters)

Quantity EquationResistance R of a uniform

conductorA

LR =

Resistors in series, sR sR = R1+ R2+ R3

Resistors in parallel, pR

321

1111

RRRRp++=

Power dissipated in resistor:

R

V

RIVIP

22

===

Potential drop across R V = I R

Dynamo Formulae

Average e.m.f. generated in each conductor =c

NpZ

60

2

WhereZ = total number of armature conductors

c = number of parallel paths through winding between positive and negative brushesWhere c = 2 (wave winding), c = 2p (lap winding)

= useful flux per pole (webers), entering or leaving the armaturep = number of pairs of poles

N = speed (revolutions per minute)

Generator Terminal volts = EG IaRa

Motor Terminal volts = EB + IaRa


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Where EG = generated e.m.f.EB = generated back e.m.f.

Ia = armature current

Ra = armature resistance

Alternating Current

RMS value of sine curve = 0.707 of maximum value

Mean Value of Sine wave = 0.637 of maximum valueForm factor = RMS value / Mean Value = 1.11

Frequency of Alternator =60

pNcycles per second

Where p is number of pairs of poles

N is the rotational speed in r/min

Slip of Induction Motor

[(Slip speed of the field Speed of the rotor) / Speed of the Field] 100

Inductors and Inductive Reactance

Physical Quantity Equation

Inductors and InductanceVL= L

td

id

Inductors in Series: LT= L1+ L2+ L3+ . . . .

Inductor in Parallel:.....

L

1

L

1

L

1

L

1

321T

+++=

Current build up(switch initially closed after having

been opened)

At t

-

L eEt)(v =

)e-E(1t)(v

t

R

=

t-

e1(R

Ei(t) = )

=R

L

Current decay(switch moved to a new position)

=t

-

o eIi(t)

vR(t) = R i(t)

vL(t) = RTi(t)


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' =TR

L

Alternating Current f = 1/T= 2 f

Complex Numbers: C = a + j b

C = M cos + j M sin 22 baM +=

=a

btan 1-

Polar form: C = M

Inductive Reactance |XL| = LCapacitive Reactance |XC| = 1 / (C)

Resistance R

Impedance Resistance: ZR= R 0Inductance: ZL= XL 90= L 90Capacitance: ZC= XC-90= 1 / (C)-90

Quantity Equation

Ohms Law for AC V = I Z

Time Domain v(t) = Vmsin (t )i(t) = Imsin (t )

Phasor Notation V = VrmsV = Vm

Components in Series ZT= Z1+ Z2+ Z3+ .

.

Voltage Divider Rule

T

xTx

Z

ZVV =

Components in Parallel...

Z

1

Z

1

Z

1

Z

1

321T

+++=


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Quantity Equation

Current Divider Rule

x

TTx

Z

ZII =

Two impedance values inparallel

21

21TZ

ZZ

ZZ

+=

Capacitance

CapacitorsC =

V

Q [F] (Farads)

Capacitor in Series.....

C

1

C

1

C

1

C

1

321T

+++=

Capacitors in Parallel .....CCCC 321T +++=

Charging a CapacitorRC

t-

eR

Ei(t) =

RC

t-

R eEt)(v =

)e-E(1t)(v RCt

-

C = = RC

Discharging aCapacitor

=t

-o e

R

Vi(t)

=t

-

oR eVt)v (

=t

-

oC eVt)v (

' = RTC

Quantity Equation

Capacitance

V

QC=


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Quantity Equation

Capacitance of a

Parallel-plate Capacitord

AC

=

d

VE =

Isolated Sphere C = 4r

Capacitors in parallel C = C1+ C2+ C3

Capacitors in series321

1111CCCC ++=

Energy stored in acharged capacitor QVCV

C

QW

2

1

2

1

2

2

2

===

If the capacitor is

isolatedC

QW

2

2

=

If the capacitor is

connected to a battery

2

2

1CVW=

For R C circuits

Charging a capacitor

Q = Qo(1 - e-t/RC);

V = Vo(1 - e-t/RC)

Discharging a capacitor Q = Qoe- t/RC

V = Voe-t/RC

If the capacitor is isolated, the presence of the dielectric decreases the potential

difference between the platesIf the capacitor is connected to a battery, the presence of the dielectric increases thecharge stored in the capacitor.

The introduction of the dielectric increases the capacitance of the capacitor


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Current in AC CircuitRMS Current

In Cartesian

form

+

=

C

LjR

CLR

VI

1

1

2

2

Amperes

In polar forms

CLR

VI

+

=

]1

[

2

2

Amperes

where

=

R

CL

s

1

tan 1

Modulus

2

2 1

+

=

CLR

VI

Amperes

Complex Impedance

In Cartesian

form

+=C

LjRZ

1

Ohms

In polar form

sC

LRZ

+=

2

2 1Ohms

Where

=

R

CL

s

1

tan 1

Modulus

=Z

2

2

1[

+ CLR ] Ohms


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Power dissipation

Average power, cosVIP= Watts

Power dissipation in a

resistor

RIP2

= Watts

Rectification

Controlled half waverectifier Average DC voltage

( )

cos12

+= mV

Volts

Controlled full waverectifier

Average DC voltage ( )

cos1 += mV

Volts

Power Factor

DC

PowerR

VRIVIPdc

22 ===

ACPower

cos).Re( VIIVPac ==

Power in ac circuits

Quantity Equation

Resistance The mean power = P = IrmsVrms= Irms2R

Inductance The instantaneous power = (Io sin wt) (Vo sin (wt +

)

The mean power P = 0

Capacitance The instantaneous power = (Io sin (wt + /2)) (Vosinwt )

The mean power P = 0

Formula for a.c.

powerThe mean power = P = IrmsVrmscos


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Three Phase Alternators

Star connected

Line voltage = VoltagePhase3 Line current = phase current

Delta connectedLine voltage = phase voltage

Line current = CurrentPhase3 Three phase power

P = Cos3 LL IE

Where:P is the active power in Watts

ELis the Line Voltage in Volts

ILis the line current in Amperes

Cos is the power factorElectrostatics

Quantity Equation

Instantaneous current,

dt

dvC

dt

dqI == Amperes

Permittivity of free space12

9

0 1085.836

10

==

Farads

(meters)-1

Energy stored in a

capacitor2

2

1CV= Joules

Quantity Equation

Coulombs law

2

21

r

QQkF=

Electric fields

q

FE=

Due to a point charge24 r

QE

o=


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Quantity Equation

Due to a conducting sphere carrying charge

Q Inside the sphere

E = 0

Outside the sphere24 r

QE

o=

Just outside a uniformly charged conductingsphere or plate o

E

=

An electric field E is a vectorThe electric field strength is directly proportional to the number of electric field linesper unit cross-sectional area,

The electric field at the surface of a conductor is perpendicular to the surface.The electric field is zero inside a conductor.

Quantity Equation

Suppose a point charge Q is at A. The work done in

bringing a charge q from infinity to some point a distancer from A is

r

QqW

o4=

Electric potential

q

WV =

Due to a point charge

r

QV

o4=

Due to a conducting sphere, of radius a, carrying charge

Q:

Inside the spherea

QV

o4=

Outside the sphere

r

QV

o4=

If the potential at a point is V, then the potential energyof a charge q at that point is

U = qV


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Quantity Equation

Work done in bringing charge q from A of potential VAto

point B of potential VB

W = q (VB VA)

Relation between E and V

dx

dVE =

For uniform electric field

d

VE =

Magnetostatics


Magnetic flux density (also called the B-

field) is defined as the force acting per unit

current length.I

FB=

Force on a current-carrying conductor in amagnetic field

F = I B F= I B And Magnitude of F= F = I Bsin

Force on a moving charged particle in amagnetic field

F = q v B

Circulating Charges

r

mvqvB

2

=

Calculation of magnetic flux density


Magnetic fields around a long straight wire

carrying current Ia

IB o

2=

where a = perp. distance from a

very long straight wire.

Magnetic fields inside a long solenoid,carrying current

I: B = on I, where n = number ofturns per unit length.

Hall effect

At equilibrium QvBd

VQ H = and VH= B v d


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The current in a material is given by I = nQAv

The forces between two current-carrying

conductorsa

IIF o

2

21

21

=


The torque on a rectangular coil in a magnetic

field

T = F b sin = N I B b sin= N I A B sin

If the coil is in a radial field and the plane of the

coil is always parallel to the field, then

T = N I A B sin = N I A B sin 90o

= N I A B

Magnetic flux = B A cos and

Flux-linkage = N

Current Sensitivity

c

NAB

ISI ==

Lenz's law

The direction of the induced e.m.f. is such that it tends tooppose the flux-change causing it, and does oppose it ifinduced current flows.

dt

d

N=

EMF Equations

E.m.f. induced in a straight conductor = B L v

E.m.f. induced between the center and the rim of a spinningdisc

= B r2f

E.m.f. induced in a rotating coil = N A B w sinwt

Quantity Equation

Self-induction

dtdIL

/

=


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Quantity Equation

N = L I

Energy stored in an inductor:

2

2

1LIU=

Transformers:

P

S

P

S

N

N

V

V=

The L R (d.c.) circuit:)1( /LRte

R

EI =

When a great load (or smaller

resistance) is connected tothe secondary coil, the flux in

the core decreases. The

e.m.f., p, in the primary coilfalls.

Vp-p = I R;R

VI

pP =

Kirchoffs laws

Kirchoff's first law (Junction Theorem)

At a junction, the total current entering the junction is equal to the total

current leaving the junction.

Kirchoff's second law (Loop Theorem)

The net e.m.f. round a circuit is equal to the sum of the p.d.s round the loop.


PowerP =

W

tVI=

Electric currentI =

q

t

Work W = q V

Ohms Law V = I R

Resistances in Series R R RT 1 2= +

Resistances in Parallel 1

R R RT 1 2= +

1 1


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F = m a

Impulse = force time = change of momentum

F t = m v m u

Newton's third law of motion

When two objects interact, they exert equal and opposite forces on one another.

"Third-law pair" of forces act on two different bodies.Universal Law

F = Gmsmp/d2

msis the mass of the sun.

mpis the mass of the planet.

The Universal law and the second law must be consistent

Newtons Laws of Motion and Their Applications

Physical Quantity Equations

Average velocity vs

t

v + u

2av = =

Acceleration a =v - u

t

Momentum p = mv

Force F = m a

Weight weight = mg

Work done W = F s

Kinetic energy E mvk2= 1

2

Gravitational potential energy E mghp=

Equations of motion a =v u

t s = ut + at v u as1

2

2 2 2 = +; ; 2

Centripetal acceleration a = vr

2

Centripetal force F = m a =mv

r

2


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Newtons Law of Universal

GravitationF = G

m m

r

1 2

2

Gravitational field strength g = G Mr2


Moment of a force M = r F

Principle of

moments

=M 0

Stress Stress FA=

StrainStrain=

ll

Youngs ModulusY

F / A=

l l/

Scalar: a property described by a magnitude only

Vector: a property described by a magnitude and a direction

Velocity: vector property equal to displacement / time

The magnitude of velocity may be referred to as speedIn SI the basic unit is m/s, in Imperial ft/s

Other common units are km/h, mi/h

Conversions:1m/s = 3.28 ft/s

1km/h = 0.621 mi/h

Speed of sound in dry air is 331 m/s at 0C and increases by about 0.61 m/s for each C

rise.

Speed of light in vacuum equals 3 x 108m/s

Acceleration:vector property equal to change in velocity time.

In SI the basic unit is m/s2


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In Imperial ft/s2

Conversion:

2228.31

s

ft

s

m=

Acceleration due to gravity, g is 9.81 m/s2

5.2.2 Linear Velocity and Acceleration

Quantity Equations

If u initial velocity and v final velocity,then displacement s,

+=2

uvs

If t is the elapsed time 2

2

1atuts +=

If a is the acceleration asuv 222 +=

Angular Velocity and Acceleration

Quantity Equations

t

+

= 221

angular displacement

(radians)angular velocity (radians/s);

1= initial, 2= final 21

2

1tt +=

angular acceleration(radians/s2)

22

1

22 +=

Linear displacement s = r

Linear velocity v = r

Linear, or tangentialacceleration

aT = r

Tangential, Centripetal and Total Acceleration

Quantity Equations


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Tangential acceleration aT is due to angular acceleration

aT = r

Centripetal (Centrifugal) acceleration ac is due to change

in direction only

ac = v2/r = r 2

Total acceleration, a, of a rotating point experiencingangular acceleration is the vector sum of aT and ac

a = aT + ac

5.2.3 Force

Vector quantity, a push or pull which changes the shape and/or motion of an object

In SI the unit of force is the newton, N, defined as a kg mIn Imperial the unit of force is the pound lb

Conversion: 9.81 N = 2.2 lb

WeightThe gravitational force of attraction between a mass, m, and the mass of the EarthIn SI weight can be calculated from Weight = F = mg, where g = 9.81 m/s

2

In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the

known weight in pounds

g

weightm=

22.32

s

ftg=

Torque Equation

T = I where T is the acceleration torque in Nm, I is the moment of inertia in kg m2andis the angular acceleration in radians/s2

Momentum

Vector quantity, symbol p,p = mv [Imperial p = (w/g)v, where w is weight]

in SI unit is kgm / s

Work

Scalar quantity, equal to the (vector) product of a force and the displacement of an

object. In simple systems, where W is work, F force and s distance

W = F sIn SI the unit of work is the joule, J, or kilojoule, kJ

1 J = 1 Nm

In Imperial the unit of work is the ft-lb

Energy

Energy is the ability to do work, the units are the same as for work; J, kJ, and ft-lb


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Kinetic Energy

22

2

1mkER=

Where k is radius of gyration, is angular velocity in rad/s

Kinetic Energy of Rotation

2

2

1IEr=

Where I = mk2is the moment of inertia

5.2.4 Centripetal (Centrifugal) Force

r

mvFc

2=

Where r is the radius

Where is angular velocity in rad/s

Potential Energy

Quantity Equation

Energy due to position in a forcefield, such as gravity

Ep = m g h

In Imperial this is usually expressed Ep = w hWhere w is weight, and h is height

above some specified datum

Thermal Energy

In SI the common units of thermal energy are J, and kJ, (and kJ/kg for specific

quantities)In Imperial, the units of thermal energy are British Thermal Units (Btu)

Conversions

1 Btu = 1055 J

1 Btu = 778 ft-lb

Electrical Energy

In SI the units of electrical energy are J, kJ and kilowatt hours kWh. In Imperial, the unit

of electrical energy is the kWh

Conversions

1 kWh = 3600 kJ

1 kWh = 3412 Btu = 2.66 x 106ft-lb


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Power

A scalar quantity, equal to the rate of doing workIn SI the unit is the Watt W (or kW)

s

JW 11 =

In Imperial, the units are:

Mechanical Power (ft lb) / s, horsepower h.p.

Thermal Power Btu / sElectrical Power - W, kW, or h.p.

Conversions

..1746 phW=

slbftph = 550..1

s

BtukW 948.01 =

Pressure

A vector quantity, force per unit area

In SI the basic units of pressure are pascals Pa and kPa

211

m

NPa=

In Imperial, the basic unit is the pound per square inch, psi

Atmospheric Pressure

At sea level atmospheric pressure equals 101.3 kPa or 14.7 psi

Pressure Conversions

1 psi = 6.895 kPa

Pressure may be expressed in standard units, or in units of static fluid head, in both SI

and Imperial systems

Common equivalencies are:1 kPa = 0.294 in. mercury = 7.5 mm mercury1 kPa = 4.02 in. water = 102 mm water1 psi = 2.03 in. mercury = 51.7 mm mercury1 psi = 27.7 in. water = 703 mm water1 m H2O = 9.81 kPa

Other pressure unit conversions:


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1 bar = 14.5 psi = 100 kPa1 kg/cm2= 98.1 kPa = 14.2 psi = 0.981 bar1 atmosphere (atm) = 101.3 kPa = 14.7 psi

Simple Harmonic Motion

Velocity of P =smxR 22

5.2.5 Stress, Strain And Modulus Of Elasticity

Youngs modulus and the breaking stress for selected materials

MaterialYoung modulus

x 1011PaBreaking stress

x 108Pa

Aluminium 0.70 2.4

Copper 1.16 4.9

Brass 0.90 4.7

Iron (wrought) 1.93 3.0

Mild steel 2.10 11.0

Glass 0.55 10

Tungsten 4.10 20

Bone 0.17 1.8

5.3 Thermodynamics5.3.1 Laws of Thermodynamics

W = PVU = Q WW= nRT lnVf/ViQ = CnTCv= 3/2RCp= 5/2RCp/Cv= = 5/3e = 1 Qc/Qh= W/Qhec= 1 Tc/ThCOP = Qc/W (refrigerators)COP = Qh /W (heat pumps)Wmax= (1-Tc/Th)QhS = Q/T


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5.3.2 Momentump = mvF = p/t

5.3.3 ImpulseI = Favt = mvf mvi

5.3.4 Elastic and Inelastic collisionmiv1i+ m2v2i= m1v1f+ m2v2f() miv1i

2+ () m2v2i

2= m1v1f

2+ m2v2f

2

miv1i+ m2v2i= (m1+ m2)vf

5.3.5 Center of Mass

xcm= mx/MVcm= mv/MAcm= ma/MMAcm= Fnet

5.3.6 Angular Motions = rvt= rat= rac= vt

2/r = r2

= 2/T

1 rev = 2rad = 360o

For constant = o+ t2= o

2+2

= ot + t2

= (o+ )t/2I = mr2KER= I

2

= rF

= IWR= L = I= IWR= L = ILi= Lf


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fbeats= | f1 f2 |Fluids= m/VP = F/AP2= P1+ gh

Patm= 1.01 x 105

Pa = 14.7 lb/in2

FB= fVg = Wf(weight of the displaced fluid)o/f= Vf /Vo (floating object)water= 1000 kg/m

3

Wa=W-FB

Equation of Continuity: Av = constant

Bernoullis equation: P + v2+ gy = 0

5.3.12 Temperature and HeatTF= (9/5) TC+32

TC= 5/9(TF-32)TF= (9/5) TCT= TC+273.15= m/vL = LoTA = AoTV = VoT =3Q = mcTQ = mL1 kcal = 4186 J

Heat Loss = Heat GainQ = (kAT)t/L,H = Q/t =(kAT)/LQ = eT4AtP = Q/tP = AeT4P net= Ae(T

4-TS

4)

= 5.67 10-8 W/m 2K4

5.3.13 Ideal GasesPV = nRT

R = 8.31 J/mol KPV = NkTNA= 6.02 10

23molecules/mol

k = 1.38 10-23J/KM=NAm(KE)av=(1/2mv

2)av= 3/2kT

U= 3/2NkT = 3/2nRT


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5.3.14 Elastic DeformationP = F/AY = FLo/AL

S = Fh/AxB = VoF / AVVolume of the sphere = 4r3/31 atm = 1.01 105Pa

5.3.15 Temperature ScalesC = 5/9 (F 32)F = (9/5) C + 32R = F + 460 (R Rankine)K = C + 273 (K Kelvin)

5.3.16 Sensible Heat EquationQ=mcTM=massC=specific heatT=temperature chance

5.3.17 Latent HeatLatent heat of fusion of ice = 335 kJ/kgLatent heat of steam from and at 100C = 2257 kJ/kg1 tonne of refrigeration = 335 000 kJ/day = 233 kJ/min

5.3.18 Gas LawsBoyles Law

When gas temperature is constant

PV = constant orP1V1= P2V2Where P is absolute pressure and V is volume

Charles Law

When gas pressure is constant,

.constT

V

= or

2

2

1

1

T

V

T

V=

where V is volume and T is absolute temperature


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Gay-Lussac's Law

When gas volume is constant,

.constT

P

=

or

2

2

1

1

T

P

T

P=

where P is absolute pressure and T is absolute temperature

General Gas Law

.2

22

1

11 constT

VP

T

VP ==

P V = m R T where P = absolute pressure (kPa)

V = volume (m3)

T = absolute temp (K)m = mass (kg)

R = characteristic constant (kJ/kgK)

AlsoPV = nRoT where P = absolute pressure (kPa)

V = volume (m3)

T = absolute temperature KN = the number of kmoles of gas

Ro = the universal gas constant 8.314 kJ/kmol/K

5.3.19 Specific Heats Of Gases

GAS

Specific Heat

at ConstantPressure

kJ/kgK or

kJ/kgo

C

Specific Heat

at ConstantVolume

kJ/kgK or

kJ/kgo

C

Ratio ofSpecific

= cp / cv

Air 1.005 0.718 1.40

Ammonia 2.060 1.561 1.32

Carbon Dioxide 0.825 0.630 1.31

Carbon 1.051 0.751 1.40


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GAS

Specific Heatat Constant

PressurekJ/kgK or

kJ/kg oC

Specific Heatat Constant

VolumekJ/kgK or

kJ/kg oC

Ratio of

Specific

= cp / cv

Monoxide

Helium 5.234 3.153 1.66

Hydrogen 14.235 10.096 1.41

Hydrogen

Sulphide1.105 0.85 1.30

Methane 2.177 1.675 1.30

Nitrogen 1.043 0.745 1.40

Oxygen 0.913 0.652 1.40

Sulphur Dioxide 0.632 0.451 1.40

5.3.20 Efficiency of Heat Engines

Carnot Cycle

1

21

T

TT =

where T1and T2are absolute temperatures of heat source and sink

Air Standard Efficiencies

Spark Ignition Gas and Oil Engines (Constant Volume Cycle)

)1(

11

=

vr

rv= compression ratio= specific heat (constant pressure) / Specific heat (constant volume)

Diesel Cycle

)1(

)1

1 1

= Rr

R

v

Where r = ratio of compression

R = ratio of cut-off volume to clearance volume

High Speed Diesel (Dual-Combustion) Cycle


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[ ])1()1(1

11 +

=

kkr

k

v

Where rv= cylinder volume / clearance volumek = absolute pressure at the end of constant V heating (combustion) / absolute pressure at

the beginning of constant V combustion

= volume at the end of constant P heating (combustion) / clearancevolume

Gas Turbines (Constant Pressure or Brayton Cycle)

=

1

11

pr

where rp = pressure ratio = compressor discharge pressure / compressor intake pressure

5.3.21 Heat Transfer by Conduction

Material Coefficient of Thermal

Conductivity

W/m C

Air 0.025

Brass 104

Concrete 0.85

Cork 0.043

Glass 1.0

Iron, cast 70

Steel 60

Wallboard,paper

0.076

Aluminum 206

Brick 0.6

Copper 380Felt 0.038

Glass, fibre 0.04

Plastic, cellular 0.04

Wood 0.15


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5.3.22 Thermal Expansion of SolidsIncrease in length = L (T2 T1)Where L = original length

= coefficient of linear expansion(T2 T1) = rise in temperature

Increase in volume = V (T2 T1)Where V = original volume

= coefficient of volumetric expansion(T2 T1) = rise in temperatureCoefficient of volumetric expansion = Coefficient of linear expansion 3

= 3

5.3.23 Chemical Heating Value of a Fuel

Chemical Heating Value MJ per kg of fuel = S

O

HC 3.9)8(1447.332

2 ++ C is the mass of carbon per kg of fuel

H2is the mass of hydrogen per kg of fuelO2is the mass of oxygen per kg of fuel

S is the mass of sulphur per kg of fuel

Theoretical Air Required to Burn Fuel

Air (kg per kg of fuel) =23

100)(8

3

822

++ SOHC

Air Supplied from Analysis of Flue Gases

Air in kg per kg of fuel = CCOCO

N

+ )(33 2

2

Boiler Formulae

Equivalent evaporationkgkj

hhms

/2257

)( 21=

Factor of evaporationkgkj

hh

/2257

)( 21=

Boiler Efficiency

)(

)( 21

aluecalorificvmf

hhms

Where

ms= mass flow rate of steam

h1= enthalpy of steam produced in boiler

h2= enthalpy of feedwater to boiler


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mf= mass flow rate of fuel


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P-V-T Relationships

Name ofprocess

Valueof n

P-V T-P T-V

Heat added Work doneChange inInternalEnergy

ConstantVolume

V=Constant --

2

1

2

1

P

P

T

T= -- ( )12 TTmcv 0 ( )12 TTmcv

Constantpressure

P=Pressure0 -- --

2

1

2

1

V

V

T

T= ( )12 TTmcp P(V2-V1) ( )12 TTmcv

IsothermalT=Constant 11

2

2

1

V

V

P

P

= -- --

2

1logP

PmRT e

2

1logP

PmRT e 0

IsentropicS=Constant

=

1

2

2

1

V

V

P

P

l

P

P

T

T

=

2

1

2

1

1

1

2

2

1

=

V

V

T

T 0 ( )21 TTmcv ( )12 TTmcv

PolytropicPVn =

Constantn

n

V

V

P

P

=

1

2

2

1

n

ln

P

P

T

T

=

2

1

2

1

1

1

2

2

1

=

n

V

V

T

T ( )12 TTmcn ( )21

1TT

n

mR

( )12 TTmcv

Thermodynamic Equations for perfect gases

*Can be used for reversible adiabatic processescv= Specific heat at constant volume, kJ/kgKcp= Specific heat at constant pressure, kJ/kgK


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cm= Specific heat for polytropic process = kgKkJn

ncv /

1

H = Enthalpy, kJ

= Isentropic Exponent, cp/cvn = polytropic exponentP = Pressure, kPa

R = Gas content, kJ/kgK

S = Entropy, kJ/KT = Absolute Temperature, K = 273+C

U = Internal Energy, kJ

V = Volume, m3

m = Mass of gas, kg


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Specific Heat and Linear

Expansion of Solids

Mean Specific Heat between 0 Co

and 100 Co kJ/kgK or kJ/kg Co

Coefficient of Lin

between 0 Co

(multiply

Aluminum 0.909 23.

Antimony 0.209 17.

Bismuth 0.125 12.

Brass 0.383 18.

Carbon 0.795 7.9

Cobalt 0.402 12.

Copper 0.388 16.

Glass 0.896 9.0

Gold 0.130 14.

Ice (between -20 Co & 0 Co ) 2.135 50.

Iron (cast) 0.544 10.

Iron (wrought) 0.465 12.

Lead 0.131 29.

Nickel 0.452 13.

Platinum 0.134 8.6

Silicon 0.741 7.8

Silver 0.235 19.

Steel (mild) 0.494 12.

Tin 0.230 26.

Zinc 0.389 16.


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Specific Heat and Volume Expansion for Liquids

Liquid

Specific Heat

(at 20 Co )

KJ/kgK or kJ/kg Co

Coefficient of Volume Expansio

(Multiply by 10-4)

Alcohal 2.470 11.0

Ammonia 0.473

Benzine 1.138 12.4

Carbon Dioxide 3.643 1.82

Mercury 0.139 1.80

Olive oil 1.633

Petroleum 2.135

Gasoline 2.093 12.0

Turpentine 1.800 9.4

Water 4.183 3.7


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5.4 Fluid Mechanics5.4.1 Discharge from an Orifice

Let A = cross-sectional area of the orifice = 2

4

d

And Ac = cross-sectional area of the jet at the venaconrtacta

2

4 cd

Then Ac = CcAOr

2

==

d

d

A

AC ccc

Where Ccis the coefficient of contraction

At the vena contracta, the volumetric flow rate Q of the fluid is given byQ = area of the jet at the vena contracta actual velocity = AcV

Or ghACCQ vc 2= Typically, values for Cd vary between 0.6 and 0.65Circular orifice: Q = 0.62 A 2ghWhere Q = flow (m3/s) A = area (m2) h = head (m)Rectangular notch: Q = 0.62 (B H) 2/3 2gh


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Where B = breadth (m)H = head (m above sill)

Triangular Right Angled Notch: Q = 2.635 H5/2

Where H = head (m above sill)

5.4.2 Bernoullis Theory

g

v

w

PhH

2

2

++=

H = total head (meters)

w = force of gravity on 1 m3of fluid (N)

h = height above datum level (meters)

v = velocity of water (meters per second)

P = pressure (N/m2or Pa)

Loss of Head in Pipes Due to Friction

Loss of head in meters =g

v

d

Lf

2

2

L = length in metersv = velocity of flow in meters per secondd = diameter in meters

f = constant value of 0.01 in large pipes to 0.02 in small pipes

5.4.3 Actual pipe dimensions

Nominal

pipe size

(in)

Outside

diameter

(mm)

Inside

diameter

(mm)

Wall

thickness

(mm)

Flow area(m2)

1/8 10.3 6.8 1.73 3.660 10-5

1/4 13.7 9.2 2.24 6717 10-5

3/8 17.1 12.5 2.31 1.236 10-4

1/2 21.3 15.8 2.77 1.960 10-4

3/4 26.7 20.9 2.87 3.437 10-4

1 33.4 26.6 3.38 5.574 10-4

1 42.2 35.1 3.56 9.653 10-4

1 48.3 40.9 3.68 1.314 10-3

2 60.3 52.5 3.91 2.168 10-3


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- 81 -

References

6.1 Periodic Table of Elements

A1

8A18

1H

1.008

2A2

3A13

4A14

5A15

6A16

7A17

2He

4.003

3Li

6.941

4Be

9.012

5B

10.81

6C

12.01

7N

14.01

8O

16.00

9F

19.00

10Ne

20.18

11Na

22.99

12Mg

24.31

3B3

4B4

5B5

6B6

7B7

8B8

8B9

8B10

1B11

2B12

13Al

26.98

14Si

28.09

15P

30.97

16S

32.07

17Cl

35.45

18Ar

39.95

19K

39.10

20Ca

40.08

21Sc

44.96

22Ti

47.90

23V

50.94

24Cr

52.00

25Mn

54.94

26Fe

55.85

27Co

58.93

28Ni

58.70

29Cu

63.55

30Zn

65.38

31Ga

69.72

32Ge

72.59

33As

74.92

34Se

78.96

35Br

79.90

36Kr

83.80

37Rb

85.47

38Sr

87.62

39Y

88.91

40Zr

91.22

41Nb

92.91

42Mo

95.94

43Tc

97.9

44Ru

101.1

45Rh

102.9

46Pd

106.4

47Ag

107.9

48Cd

112.4

49In

114.8

50Sn

118.7

51Sb

121.8

52Te

127.6

53I

126.9

54Xe

131.3

55

Cs132.

9

56

Ba137.

3

57

La138.

9

72

Hf178.

5

73

Ta180.

9

74

W183.

8

75

Re186.

2

76

Os190.

2

77

Ir192.

2

78

Pt195.

1

79

Au197.

0

80

Hg200.

6

81

Tl204.

4

82

Pb207.

2

83

Bi209.

0

84

Po(209)

85

At(210)

86

Rn(222)

87Fr

(223)

88Ra

226.0

89Ac

227.0

104Rf

(261)

105Db

(262)

106Sg

(266)

107Bh

(264)

108Hs

(265)

109Mt

(268)

58Ce

140.1

59Pr

140.9

60Nd

144.2

61Pm

(145)

62Sm150.

4

63Eu

152.0

64Gd

157.3

65Tb

158.9

66Dy

162.5

67Ho

164.9

68Er

167.3

69Tm168.

9

70Yb

173.0

71Lu

175.0

90

Th232.

0

91

Pa231.

0

92

U238.

0

93

Np237.

0

94

Pu(244)

95

Am(243)

96

Cm(247)

97

Bk(247)

98

Cf(251)

99

Es(252)

100

Fm(257)

101

Md(258)

102

No(259)

103

Lr(262)

Chapter 6


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6.2 Resistor Color Coding

Color Value

Black 0

Brown 1

Red 2

Orange 3

Yellow 4

Green 5

Blue 6

Violet / Purple 7

Grey 8

White 9

Courtesy: Dick Smith Electronics, Australia


89/99

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Practical Shielding, EMC/EMI, Noise Reduction, Earthing and Circuit Board Layout of

Electronic Systems

Practical Power Electronics & Switch Mode Power Supply Design for IndustryInformation Technology

Industrial Network Security for SCADA, Automation, Process Control and PLC Systems

Practical Web-Site Development & E-Commerce Systems for Industry

Chemical Engineering

Practical Fundamentals of Chemical Engineering

Instrumentation, Automation & Process Control

Practical Analytical Instrumentation in On-Line Applications

Practical Alarm Systems Management for Engineers and Technicians

Troubleshooting Programmable Logic Controller's for Automation and Process Control

Practical Batch Management & Control (Including S88) for Industry

Practical Boiler Control and Instrumentation for Engineers and Technicians

Practical Programming for Industrial Control - using ( IEC 1131-3 and OPC )

Practical Troubleshooting of Data Acquisition & SCADA Systems for Engineers and

Technicians

Practical Industrial Flow Measurement for Engineers and Technicians

Practical Hazops, Trips and Alarms

Practical Hazardous Areas for Engineers and Technicians

A Practical Mini MBA in Instrumentation and Automation

Practical Instrumentation for Automation and Process Control

Practical Intrinsic Safety for Engineers and TechniciansPractical Tuning of Industrial Control Loops

Practical Motion Control for Engineers and Technicians

Practical Fundamentals of OPC (OLE for Process Control)

Practical Process Control for Engineers and Technicians

Practical Process Control & Tuning of Industrial Control Loops

Practical SCADA & Telemetry Systems for Industry

Practical Shutdown & Turnaround Management for Engineers and Managers


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Practical Safety Instrumentation & Emergency Shutdown Systems for Process Industries

using IEC 61511 and IEC 61508Practical Fundamentals of E-Manufacturing, Manufacturing Execution Systems (MES)and Supply Chain Management

Practical Industrial Programming using 61131-3 for Programmable Logic Controllers(PLCs)

Control Valve Sizing, Selection and Maintenance

Best Practice in Process, Electrical and Instrumentation Drawings & Documentation

Practical Distributed Control Systems (DCS)

Mechanical Engineering

Practical Fundamentals of Heating, Ventilation & Air-conditioning (HVAC) for

Engineers & Technicians

Practical Boiler Plant Operation and Management for Engineers and Technicians

Practical Cleanroom Technology and Facilities for Engineers and Technicians

Practical Hydraulic Systems: Operation and Troubleshooting

Practical Lubrication Engineering for Engineers and Technicians

Practical Safe Lifting Practice and Maintenance

Practical Centrifugal Pumps - Optimizing Performance

Practical Machinery and Automation Safety for Industry

Practical Machinery Vibration Analysis and Predictive Maintenance

Practical Pneumatics:

engineering formulas book

Documents